top95 X-cycle (simple coloring), Y-cycle, and "3D-Medusa" examples

This document discusses three types of "logical chains" that people employ in
solving Sudoku Puzzles. These include X-cycles, Y-cycles, and 3D Medusa.
See also top95-methods.htmexplain.htm.
See examples.htm for almost 1600 examples categorized by type culled from the top95.
X-CyclesX-cyclesY-cycles3D Medusa
--------
X-Cycle analysis (Simple coloring) just involves following "conjugate pairs"
of a specific number around to see if they force a possibility to be eliminated.
"Conjugate pair" here refers to exactly two occurances of a specific number
("candidate") in a specific row, column, or block. In the diagrams shown below,
conjugate pairs are shown using "blades"---dashed lines that mean there are NO
other occurances of this number in a row (--), column (|), or block (\ or /).
While there are certainly many examples of this sort of thing in the
top95 collection, there are just four puzzles in top95 that can be solved using
only basic methods (cross-hatch scanning, locked candidates, and subset analysis)
plus simple coloring (X-cycle) analysis. These include
#5, #13, #17, #24, #25, #38, #41, #47, #53, #63, #71, and #95.
Only a few examples are shown below.
The figures below use the notation introduced in the Sudoku Programmer Forum
Click on the puzzle number to load the original configuration.
Click on the description to load the exact configuration.
Puzzle #13weak 3-blade fishy cycle eliminates r4c2#5:
r4c2#5 |
x......5*
. |
. |
-5------+--5*-
| /
|/
5
|
weak 3-blade pentagon eliminates r5c6#5:
| r5c6#5
5*.......x
| .
| .
| .
| 5
| /
| /
-5-----5*-
|
weak 2-blade pentagon eliminates r4c2#7:
r4c2#7 |
x....7*
| . |
|. |
7 |
| |
| |
| |
7*......7
| |
weak 2-blade fishy cycle eliminates r5c5#7:
|
7*
| | .
| | .
7...........x r5c5#7
| |
| |
| |
7*......7
| |
Puzzle #25strong 4-blade fishy cycle on 5 eliminates all with *:
-1----------1*-
| |
| |
1*...........1*
| |/
1
|
Puzzle #53strong 6-blade fishy cycle on 7 eliminates all with *:
| |
-7*-----7-
| |
| | |
-7------+-------7*-
| | |
| |
| |
| |
| 7
| /|
| /
7*...7*
strong 4-blade pentagon eliminates all with *:
|
-5--------5*-
| /
| |/
| 5 (more cycles exist in this example)
| |
| |
5*....5*
| |
Puzzle #71weak 3-blade pentagon eliminates r4c3#7:
-7*-------7-
. /
. |/
. 7*
. |
. |
x.....7
r4c3#7 |
Y-CyclesX-cyclesY-cycles3D Medusa
--------
I learned this term from Glenn Fowler. The idea is that cells
with exactly two candidates can be traversed one to the
next to the next, because in each pair if one value is "FALSE"
then the other will be "TRUE". We can treat them just like
conjugate pairs, except we must switch numbers each step.
It's really very easy to do. In the language of strong chains,
each of the cells involved in a Y cycle is a short strong chain,
and each connection to the next cell is a weak link. (It might
be strong -- a conjugate pair -- but that is not required.)
What's nice about this is that nothing else on the board matters
at all for the analysis -- just the cells with exactly two
possible values.
The examples from top95 that can be solved using only basic methods plus
Y-cycle include #4, #5, #17, #41, #74, and #76.
Several examples are shown below.
Click on the puzzle number to load the original configuration.
Click on the description to load the exact configuration.
Puzzle #4
a three-cell Y cycle allowing for the elimination of r2c3#9:
r2c3#9
9...x
5 .
. .
5 .
8. .
. 9
8
a four-cell Y cycle allowing for the elimination of r4c2#4:
r4c2#4
x..4
. 9
. .
. .
4 .
5...5 .
8 .
. .
. .
. .
. 9
8
a four-cell Y cycle allowing for the elimination of r2c2#5:
r2c2#5
5.....x
9 .
. .
. .
. .
9 .
6 .
. .
. .
6 .
4 .
. .
. .
. 5
./
4
#5a four-cell Y cycle allowing the elimination of r9c1#9:
5..............5
9 9
. .
. .
. .
. .
. .
x.........9 .
r9c1#9 5 .
. .
. 9
5
a four-cell Y cycle allowing the elimination of r9C1#5:
9............9
5 5
. .
. .
. .
. .
. .
x.........5 .
r9c1#5 9 .
. .
. 5
9
The above two, though independent, can be seen together
as a double Y cycle involving only 5 and 9.
Puzzle #17another double 4-cell Y cycle allowing the elimination of both r1c3#4 and r1c3#6:
r1c3#4,6
x.......4
. x.....6.
. . ..
4 . ..
.6 ..
.. ..
.. ..
.6........6.
4..........4
The position of the deletion is set because that is the only cell
that has any possibilities other than 4 and 6.
The actual logic, remember, involves BOTH 4 and 6 and goes
-46...64...46...64- and -64...46...64...46-
FT-->FT-->FT-->FT FT-->FT-->FT-->FT
TFT->F->T F->T->F->T
Tgoing either way.
3D MedusaX-cyclesY-cycles3D Medusa
---------
As useful as X-cycles and Y-cycles are, one could imagine an improvement
if they were combined. We could "transfer" the strongness of a two-valued
cell to an associated conjugate pair. This synthesis of X and Y I've chosen
to call "3D Medusa" because looking at the 3D rendition at the Sudoku Assistant
and the way the influence of a cell branches out all over the volume made me
think of Medusa and her killer hair. Here are the facts: Using X-cycles alone
we can solve 12 more than the basic 35 puzzles of the top95 collection. Using
Y-cycles alone, we can solve only six more than the basic 35, and only two of
those (#74, and #76) are ones that X-cycle analysis can't solve.
But, if we allow chains to combine both X and Y characteristics, then we can
solve an additional 13 puzzles of the collection that neither X-cycle or
Y-cycle analysis together can handle. In fact, there is only one puzzle, #85,
in the top95 collection that is solvable without trial and error and actually
requires something more than 3D Medusa alone. This one has an intractable
X-wing. (Several puzzles have X-wings or swordfish, but 3D Medusa is able to solve
these puzzles without need for finding them.)
OK, so the set of additional puzzles that 3D Medusa cracks but X-cycle and Y-cycle
analysis alone can't handle include the following:
#8, #10, #29, #32, #52, #55, #70, #72, #77, #80, #82, #87, and #94.
Shown here are a few examples. They all come from configurations that have no
avenues of advancement involving pure X-cycles, Y-cycles, X-Wings, or Swordfish.
(Go ahead and look.... I guarantee you won't find any. It is possible, however,
that some of these solutions might benefit from uniqueness tests. The Sudoku
Assistant does not use uniqueness tests. I do when I'm hand-solving, though,
because they're so handy.)
As a first example of using the 3D Medusa strategy, consider puzzle #8.
In that puzzle we find this configuration.
Shown below is the 3D Medusa "strong chain analysis" for this configuration.
The letters are chain designations. Nineteen strong chains
are labeled A-S, with alternating parity shown using lowercase.